20731_Problem_solving_Year_3_Number_and_Place_Value_2

Your partner in education 

YEAR 3 

PROBLEM-SOLVING 

IN MATHEMATICS 

Number and place value – 2 

Two-time winner of the Australian 

Primary Publisher of the Year Award

Problem-solving in mathematics 

(Book D) 

Published by R.I.C. Publications ® 2008 

Copyright © George Booker and 

Denise Bond 2007 

RIC–20731 

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FOREWORD 

Books A–G of Problem-solving in mathematics have been developed to provide a rich resource for teachers 

of students from the early years to the end of middle school and into secondary school. The series of problems, 

discussions of ways to understand what is being asked and means of obtaining solutions have been built up to 

improve the problem-solving performance and persistence of all students. It is a fundamental belief of the authors 

that it is critical that students and teachers engage with a few complex problems over an extended period rather than 

spend a short time on many straightforward ‘problems’ or exercises. In particular, it is essential to allow students 

time to review and discuss what is required in the problem-solving process before moving to another and different 

problem. This book includes extensive ideas for extending problems and solution strategies to assist teachers in 

implementing this vital aspect of mathematics in their classrooms. Also, the problems have been constructed and 

selected over many years’ experience with students at all levels of mathematical talent and persistence, as well as 

in discussions with teachers in classrooms, professional learning and university settings. 

Problem-solving does not come easily to most people, 

so learners need many experiences engaging with 

problems if they are to develop this crucial ability. As 

they grapple with problem, meaning and find solutions, 

students will learn a great deal about mathematics 

and mathematical reasoning; for instance, how to 

organise information to uncover meanings and allow 

connections among the various facets of a problem 

to become more apparent, leading to a focus on 

organising what needs to be done rather than simply 

looking to apply one or more strategies. In turn, this 

extended thinking will help students make informed 

choices about events that impact on their lives and to 

interpret and respond to the decisions made by others 

at school, in everyday life and in further study. 

Student and teacher pages 

The student pages present problems chosen with a 

particular problem-solving focus and draw on a range 

of mathematical understandings and processes. 

For each set of related problems, teacher notes and 

discussion are provided, as well as indications of 

how particular problems can be examined and solved. 

Answers to the more straightforward problems and 

detailed solutions to the more complex problems 

ensure appropriate explanations, the use of the 

pages, foster discussion among students and suggest 

ways in which problems can be extended. Related 

problems occur on one or more pages that extend the 

problem’s ideas, the solution processes and students’ 

understanding of the range of ways to come to terms 

with what problems are asking. 

At the top of each teacher page, there is a statement 

that highlights the particular thinking that the 

problems will demand, together with an indication 

of the mathematics that might be needed and a list 

of materials that could be used in seeking a solution. 

A particular focus for the page or set of three pages 

of problems then expands on these aspects. Each 

book is organised so that when a problem requires 

complicated strategic thinking, two or three problems 

occur on one page (supported by a teacher page with 

detailed discussion) to encourage students to find 

a solution together with a range of means that can 

be followed. More often, problems are grouped as a 

series of three interrelated pages where the level of 

complexity gradually increases, while the associated 

teacher page examines one or two of the problems in 

depth and highlights how the other problems might be 

solved in a similar manner. 

R.I.C. Publications ® www.ricpublications.com.au Problem-solving in mathematics 

iii

FOREWORD 

Each teacher page concludes with two further aspects 

critical to successful teaching of problem-solving. A 

section on likely difficulties points to reasoning and 

content inadequacies that experience has shown may 

well impede students’ success. In this way, teachers 

can be on the look out for difficulties and be prepared 

to guide students past these potential pitfalls. The 

final section suggests extensions to the problems to 

enable teachers to provide several related experiences 

with problems of these kinds in order to build a rich 

array of experiences with particular solution methods; 

for example, the numbers, shapes or measurements 

in the original problems might change but leave the 

means to a solution essentially the same, or the 

context may change while the numbers, shapes or 

measurements remain the same. Then numbers, 

shapes or measurements and the context could be 

changed to see how the students handle situations 

that appear different but are essentially the same 

as those already met and solved. Other suggestions 

ask students to make and pose their own problems, 

investigate and present background to the problems 

or topics to the class, or consider solutions at a more 

general level (possibly involving verbal descriptions 

and eventually pictorial or symbolic arguments). 

In this way, not only are students’ ways of thinking 

extended but the problems written on one page are 

used to produce several more problems that utilise 

the same approach. 

Mathematics and language 

The difficulty of the mathematics gradually increases 

over the series, largely in line with what is taught 

at the various year levels, although problem-solving 

both challenges at the point of the mathematics 

that is being learned as well as provides insights 

and motivation for what might be learned next. For 

example, the computation required gradually builds 

from additive thinking, using addition and subtraction 

separately and together, to multiplicative thinking, 

where multiplication and division are connected 

conceptions. More complex interactions of these 

operations build up over the series as the operations 

are used to both come to terms with problems’ 

meanings and to achieve solutions. Similarly, twodimensional 

geometry is used at first but extended 

to more complex uses over the range of problems, 

then joined by interaction with three-dimensional 

ideas. Measurement, including chance and data, also 

extends over the series from length to perimeter, and 

from area to surface area and volume, drawing on 

the relationships among these concepts to organise 

solutions as well as giving an understanding of the 

metric system. Time concepts range from interpreting 

timetables using 12-hour and 24-hour clocks while 

investigations related to mass rely on both the concept 

itself and practical measurements. 

The language in which the problems are expressed is 

relatively straightforward, although this too increases 

in complexity and length of expression across the books 

in terms of both the context in which the problems 

are set and the mathematical content that is required. 

It will always be a challenge for some students 

to ‘unpack’ the meaning from a worded problem, 

particularly as problems’ context, information and 

meanings expand. This ability is fundamental to the 

nature of mathematical problem-solving and needs to 

be built up with time and experiences rather than be 

iv 

Problem-solving in mathematics www.ricpublications.com.au R.I.C. Publications ®

FOREWORD 

diminished or left out of the problems’ situations. One 

reason for the suggestion that students work in groups 

is to allow them to share and assist each other with 

the tasks of discerning meanings and ways to tackle 

the ideas in complex problems through discussion, 

rather than simply leaping into the first ideas that 

come to mind (leaving the full extent of the problem 

unrealised). 

An approach to solving problems 

Try 

an approach 

Explore 

means to a solution 

Analyse 

the problem 

The careful, gradual development of an ability to 

analyse problems for meaning, organising information 

to make it meaningful and to make the connections 

among them more meaningful in order to suggest 

a way forward to a solution is fundamental to the 

approach taken with this series, from the first book 

to the last. At first, materials are used explicitly to 

aid these meanings and connections; however, in 

time they give way to diagrams, tables and symbols 

as understanding and experience of solving complex, 

engaging problems increases. As the problem forms 

expand, the range of methods to solve problems 

is carefully extended, not only to allow students to 

successfully solve the many types of problems, but 

also to give them a repertoire of solution processes 

that they can consider and draw on when new 

situations are encountered. In turn, this allows them 

to explore one or other of these approaches to see 

whether each might furnish a likely result. In this way, 

when they try a particular method to solve a new 

problem, experience and analysis of the particular 

situation assists them to develop a full solution. 

Not only is this model for the problem-solving process 

helpful in solving problems, it also provides a basis for 

students to discuss their progress and solutions and 

determine whether or not they have fully answered 

a question. At the same time, it guides teacher 

questions of students and provides a means of seeing 

underlying mathematical difficulties and ways in 

which problems can be adapted to suit particular 

needs and extensions. Above all, it provides a common 

framework for discussions between a teacher and 

group or whole class to focus on the problem-solving 

process rather than simply on the solution of particular 

problems. Indeed, as Alan Schoenfeld, in Steen L (Ed) 

Mathematics and democracy (2001), states so well, in 

problem-solving: 

getting the answer is only the beginning rather than 

the end … an ability to communicate thinking is 

equally important. 

We wish all teachers and students who use these 

books success in fostering engagement with problemsolving 

and building a greater capacity to come to 

terms with and solve mathematical problems at all 

levels. 

George Booker and Denise Bond 


v

CONTENTS 

Foreword .................................................................. iii – v 

Contents .......................................................................... vi 

Introduction ........................................................... vii – xix 

A note on calculator use ................................................ xx 

Teacher notes.................................................................... 2 

Stacking shapes................................................................ 3 

Painted cubes.................................................................... 4 

Cube painting.................................................................... 5 

Teacher notes.................................................................... 6 

How many digits?............................................................. 7 

Teacher notes.................................................................... 8 

Star gaze........................................................................... 9 

Magic squares/Famous magic squares.......................... 10 

Sudoku............................................................................ 11 

Teacher notes.................................................................. 12 

The water park................................................................ 13 

Lily pads and frogs.......................................................... 14 

Gone fishing.................................................................... 15 

Teacher notes.................................................................. 16 

Chicken takeaway........................................................... 17 

Teacher notes.................................................................. 18 

The big race.................................................................... 19 

Serial numbers............................................................... 20 

Puzzle scrolls................................................................... 21 

Teacher notes.................................................................. 22 

Samantha’s flower shop.................................................. 23 

Herb market.................................................................... 24 

At the bakery................................................................... 25 

Teacher notes.................................................................. 26 

Numbers in columns....................................................... 27 

Teacher notes.................................................................. 28 

Calculator problems........................................................ 29 

Ice-cream cones.............................................................. 30 

Chocolate frogs............................................................... 31 

Teacher notes.................................................................. 32 

How many?..................................................................... 33 

How far?.......................................................................... 34 

How much?..................................................................... 35 

Teacher notes.................................................................. 36 

Tropical Cairns................................................................. 37 

Teacher notes ................................................................. 38 

Squares and perimeters.................................................. 39 

More perimeters............................................................. 40 

Perimeters in squares.................................................... 41 

Teacher notes.................................................................. 42 

Beading........................................................................... 43 

Library............................................................................. 44 

At the station.................................................................. 45 

Teacher notes.................................................................. 46 

Bookworms..................................................................... 47 

Teacher notes ................................................................. 48 

Farm trails ...................................................................... 49 

Balancing........................................................................ 50 

Squares and area............................................................ 51 

Teacher notes.................................................................. 52 

DVD rentals..................................................................... 53 

Drive time........................................................................ 54 

Clock watching................................................................ 55 

Teacher notes.................................................................. 56 

On the farm..................................................................... 57 

In the barn....................................................................... 58 

Market day...................................................................... 59 

Teacher notes.................................................................. 60 

Shopping......................................................................... 61 

Solutions ..................................................................62–66 

Isometric resource page .............................................. 67 

0–99 board resource page ............................................ 68 

4-digit number expander resource page (x 3) ............... 69 

10 mm x 10 mm grid resource page ............................. 70 

15 mm x 15 mm grid resource page ............................. 71 

Triangular grid resource page ...................................... 72 

vi 


TEACHER NOTES 

Problem-solving objective 

To analyse and use information in word problems. 

Materials 

place value chart or a calculator 

Focus 

These pages explore word problems that mostly require 

addition or subtraction. Students need to determine what 

the problem is asking and, in many cases, calculate more 

than one step in order to find solutions. Analysis of the 

problems reveals that some questions contain additional 

information that is not needed. 

If necessary, materials can be used to assist with the 

calculation as these problems are about reading for 

information and determining what the problem is asking 

rather than computation or basic facts. 

Discussion 

Page 13 

These problems involve more than one step and may 

involve addition as well as subtraction. The wording 

has been kept simple to assist with the problem-solving 

process. 

Students may choose a number of different ways to find 

a solution. For example, the second problem about people 

getting out (43) could be subtracted from the people going 

swimming (79) and then this number (36) could be added 

to 397 or, alternatively, 79 could be added to 397 and then 

43 subtracted to obtain a solution. Students should be 

encouraged to explore and try different ways of arriving 

at a solution. 

Page 15 

A careful reading of each problem is needed to determine 

what the question is asking. In some cases, there is more 

information than needed and some problems contain 

numbers that are not needed to find a solution. Most 

problems require more than one step and both addition 

and subtraction are needed at times. 

Again, there are a number of ways to find a solution 

and students should be encouraged to explore and try 

different possibilities of arriving at an answer. The last 

question involves students understanding that perch and 

carp are varieties of fish, while yabbies are crustaceans. 

Possible difficulties 

• Inability to identify the need to add, subtract or 

multiply 

• Confusion over the need to carry out more than one 

step to arrive at a solution 

• Using all the numbers listed in the problems rather 

than just the numbers needed 

Extension 

• Explore the possibilities as to whether the dry season 

would be before or after spring. 

• Discuss how some problems can have more than one 

answer depending on different interpretations. 

• Students could write their own problems and give 

them to others to solve. 

Page 14 

This investigation relates to information about a lake with 

lily pads and frogs. The scenario begins with a certain 

number of frogs and lily pads. As new information is 

introduced, the numbers change to meet the new criteria; 

lily pads flower, grow and die while the frogs move from 

one lake to another. 

Students are required to keep track of the new information 

and use it to answer the subsequent questions. The 

last question has two possible answers as it depends 

on whether spring is before or after the dry season. 

Students should be encouraged to explore the different 

possibilities. 

12 


THE WATER PARK 

1. 361 adults and 173 children go 

through the gates before lunch, 

and 219 adults and 106 children 

enter after lunch. How many 

more adults than children are 

there? 

2. A total of 397 people are 

swimming in the six pools. 

Another 79 people go swimming 

while 43 people get out. How 

many people are now swimming 

in the pools? 

3. 248 people are lying on their 

towels. Later, 78 people go 

swimming, 26 people go for a 

walk and 36 people leave and 

get something to eat. How many 

people are still lying on their 

towels? 

4. In the water, 93 people are floating 

on swimming mats, 134 are 

swimming and 83 are wading in 

the shallow water. Soon, another 

21 people with swimming mats 

arrive, but 14 also get out. How 

many people are now floating on 

swimming mats? 

5. At the cafeteria, 143 people are 

sitting eating lunch and 31 are 

standing in line waiting to order 

lunch. Two large tables of 12 finish 

their lunch and leave. How many 

people are now in the cafeteria? 

6. In the wave pool, 73 surfers are 

waiting to catch a wave. A large 

wave comes and 36 surfers catch 

and ride it to the beach. How 

many did not catch the wave? 


13

LILY PADS AND FROGS 

A lake has 279 lily pads and 372 frogs. 

1. 87 of the lily pads are in flower. If each lily pad has three flowers, how 

many flowers are there altogether? 

2. How many lily pads are not in flower? 

3. Frogs like to sleep under the lily flowers. How many frogs can not sleep 

under a flower? 

4. When the rains come, 38 more lily pads burst into flower. How many lily 

pads are now in flower? 

5. Are there now enough lily flowers for each frog to sleep under? 

6. During spring, 129 new lily pads grow, 186 tadpoles turn into frogs and 

75 lily pads die. How many lily pads are now in the lake? 

7. During the dry season, some frogs move to another lake. If 148 frogs 

move to another lake, how many stay behind? 

14 


GONE FISHING 

1. On Saturday, 97 large yabbies, 21 small yabbies and 57 carp were 

caught. On Sunday, 126 large yabbies, 34 small yabbies and 83 carp 

were caught. How many yabbies were caught? 

2. During the week, 163 perch, 394 carp and 304 bass were caught in the 

dam. How much more bass than perch was caught? 

3. During a fishing competition, 923 perch were caught, tagged and then 

released back into the dam. 359 tagged perch were caught in October, 

271 in November and 106 in December. How many tagged perch have 

not been caught? 

During the first weekend in June, 239 perch, 56 large yabbies, 17 small 

yabbies and 43 carp were caught. The next weekend, 161 perch, 79 large 

yabbies, 24 small yabbies and 65 carp were caught. 

4. During which weekend were the most yabbies caught? 

5. During which weekend were the least amount of fish and yabbies 

caught? 

6. During which weekend were the most fish caught? 


15

SOLUTIONS 

Note: Many solutions are written statements rather than just numbers. This is to encourage teachers and students to solve 

problems in this way. 

THE WATER PARK..............................................Page 13 

1. 301 more adults 

2. 433 people swimming in the pools 

3. 108 people lying on their towels 

4. 100 people floating on swimming mats 

5. 150 people 

6. 37 surfers did not catch a wave. 

LILY PADS AND FROGS ....................................Page 14 

1. 261 flowers 

2. 192 lily pads 

3. 111 frogs 

4. 125 lily pads are in flower 

5. Yes, there are 375 flowers 

6. 333 lily pads 

7. If the dry season is before spring, 224 frogs stayed 

behind. If the dry season is after spring, 410 frogs stayed 

behind. 

GONE FISHING ..................................................Page 15 

1. 278 yabbies were caught. 

2. 141 more bass were caught. 

3. 187 perch have not been caught. 

4. Second weekend 

5. First weekend 

6. First weekend: 282 fish were caught—yabbies are not 

fish. 


63

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